One of the many problems I have had in my 06WK is that when turning left slowly, I feel a strong 'thud' with a jerk from what seemed like the transmission, as if it skipped a gear. The jerk is quite strong, and at some point I can hear a soft 'whirring' noise which comes and goes as I adjust the steering (but only when the wheel is pointing to the left). When turning left at high speed, there is no jerk but the whirring noise can sometimes still be heard amidst the engine.

Another new problem is that my fuel consumption has increased. I have just changed new tyres (original size), changed engine oil, air filter, serviced A/C. It feels like the car is not rolling as smoothly as before, in fact no longer moving without accelerating if there is a slight gradient on the road.

My mechanic can't seem to figure out the cause of both problems above. Can anybody help please? Thanks!!

It is 4.7L AWD.
The 'jerk' happens only when turning left while driving forward, but smooth in reversing. Several months ago my mechanic changed the transmission oil sump pan because it was leaking around the edges (apparently the previous owner had some sort of accident and did not replace it). I am worried it might be a transmission problem.

You need to get the Jeep on a two poster hoist and find the issue pronto. If need be change your mechanic because evidently he's run out of talent.
I'd suggest running the vehicle with 4 wheels off the ground in gear, turning the steering wheel etc until you find the issue.
You may have a differential problem or it could even be an ABS/stability control failure.
I wish I could help you more mate but I'm way across the pond.